Communication in bounded depth circuits


We show that rigidity of matrices can be used to prove lower bounds on depth 2 circuits and communication graphs. We prove a general nonlinear bound on a certain type of circuits and thus, in particular, we determine the asymptotic size of depthd superconcentrators for all depths ≥4 (for even depths ≥4 it has been determined before).

This is a preview of subscription content, access via your institution.


  1. [1]

    N. Alon, andP. Pudlák: Superconcentrators of depth 2 and 3,Journ. of Computer and System Science 48(1) (1994), 194–202.

    Google Scholar 

  2. [2]

    D. Dolev, C. Dwork, N. Pippenger andA. Wigderson: Superconcentrators, generalizers and generalized connectors (preliminary version),Proc. ACM STOC (1983), 42–51.

  3. [3]

    N. Pippenger: Superconcentrators of depth 2,J. Comput. System Sci. 24 (1982), 82–90.

    Google Scholar 

  4. [4]

    N. Pippenger: Communications networks, inHandbook of Theoretical Computer Science, Ed. J. van Leeuwen, Elsevier, (1990), 806–833.

  5. [5]

    N. Pippenger andA. C.-C. Yao: Rearrangeable networks with limited depth,SIAM J. Alg. Disc. Meth. 3 (1981), 411–417.

    Google Scholar 

  6. [6]

    P. Pudlák andP. Savický: On shifting networks,Theoretical Computer Science 116 (1993), 415–419.

    Google Scholar 

  7. [7]

    P. Pudlák andZ. Vavřín: Computation of rigidity of ordern 2/r for one simple matrix,Comment. Math. Univ. Carolinae 32(2) (1991), 213–218.

    Google Scholar 

  8. [8]

    A. A. Razborov: On rigid matrices (in Russian), unpublished.

  9. [9]

    V. Shoup andR. Smolensky: Lower bounds for polynomial evaluation and interpolation,Proc. IEEE FOCS (1991), 378–383.

  10. [10]

    L. G. Valiant: Graph-theoretic arguments in low level complexity,Proc. MFCS 1977, Springer-Verlag LNCS, (1977), 162–176.

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Pudlák, P. Communication in bounded depth circuits. Combinatorica 14, 203–216 (1994).

Download citation

AMS subject classification code (1991)

  • 68 R 10
  • 05 C 35